Hi there!
I understand that you would like to use PayPal as part of Square, so you can have a single inventory and avoid issues with multiple inventories.
After checking further, I see that Square indeed seems to have this feature natively, but it’s not something that we have as part of the WooCommerce extension, and I’m not even sure if that’s something that they would allow on their API.
Still, I believe that you wouldn’t need to do that as you mentioned that your goal is to integrate the inventory.
WooCommerce has its own inventory and the Square extension would allow to get the WooCommerce products inventory and integrate with Square.
You can see more about that here:
https://docs.woocommerce.com/document/woocommerce-square/#section-7
In this case, you would be able to manage fully from WooCommerce and all the new purchases on PayPal would update the inventory everywhere.
With that in mind, I would recommend you to add PayPal Payments as an independent plugin.
Feel free to get back to us anytime if you need further help!